Question

what is the ratio between root mean square speed average speed and most probable speed of the molecules in the sample of the gas

Answer 1

According to Kinetic Molecular Theory, gaseous particles are in a state of constant random motion; individual particles move at different speeds, constantly colliding and changing directions.  

Root-mean-square speed  

The root-mean-square speed is the measure of the speed of particles in a gas, defined as the square root of the average velocity-squared of the molecules in a gas.

Most probable speed

Most probable speed is the one which informs about the speed possessed by maximum number of molecules of the gas.  

The ratio

Vrms         :      Vavs           :   Vmps              

√(3RT/M)  ∶   √(8RT/πM)    : √(2RT/M)  

1.732         :      1.596          :    1.414          

1.224         :       1.128         :     1

Answer 2

Answer : The ratio between root mean square speed average speed and most probable speed of the molecules in the sample of the gas is, 1.224 : 1.128 : 1

Explanation :

Root-mean-square speed, average speed and most probable speed : It measures the speed of particles in a gas. It is defined as the speed of particle is directly proportional to the square-root of temperature of gas and inversely proportional to the square-root of molar mass of gas.

The formula of root-mean-square speed, average speed and most probable speed are :

$\nu_{rms}=\sqrt{\frac{3RT}{M}}\\\\\nu_{av}=\sqrt{\frac{8RT}{\pi M}}\\\\\nu_{mps}=\sqrt{\frac{2RT}{M}}$

The ratio between the root-mean-square speed, average speed and most probable speed will be,

$\nu_{rms}:\nu_{av}:\nu_{mps}\\\\\sqrt{\frac{3RT}{M}}:\sqrt{\frac{8RT}{\pi M}}:\sqrt{\frac{2RT}{M}}\\\\\sqrt{3}:\sqrt{\frac{8}{\pi}}:\sqrt{2}\\\\1.732:1.596:1.414\\\\1.224:1.128:1$

Therefore, the ratio between root mean square speed average speed and most probable speed of the molecules in the sample of the gas is, 1.224 : 1.128 : 1